Is this a window unit or central system?

It could be a couple of problems but my guess is, the motor gets hot while running and when it shuts off, the residual heat causes the internal thermal overload to open. When it cools back down, the overload closes and the fan runs. The problem is either with the fan itself or a component in the fan circuit.

There's a couple of things that could cause this. Without being there and hearing and seeing the issue, my first thought is the vents around the motor could be clogged with dust, preventing air flow over the motor. I've also seen mouse nests in them blocking air and adding resistance to the motor as it turns. If there is not enough air flow over the fan, it will heat up. Is your air filter REALLY dirty? You don't mention a squealing noise but resistance in the bearings could cause this. It could also be a capacitor. If the blower uses a start capacitor and that has gone, you would get the amperage draw you describe.

Then it could just be the fan has lived it's life and needs to be replaced.

Thank you for the response. It is a central air unit. The starter capacitor was replaced this past winter. The in-house filter has been cleaned - is there another filter in the air handler unit that should be cleaned?

The outside condenser fan works fine even when the inside fan is not working.

I suspect from what you describe and that if it's not the capacitor, it's time to replace the motor. There's a remote possibility the fan relay is malfunctioning but from what you describe, I stick with a bad motor. Even if it's not overheating but the internal thermal overload in the motor is malfunctioning, the motor has to be replaced.